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## PowerPoint Slideshow about ' Graphical Examination of Data' - pembroke

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Presentation Transcript

Sources

- H. Anderson, T. Black: Multivariate Data Analysis,(5th ed., p.40-46).
- Yi-tzuu Chien: Interactive Pattern Recognition,(Chapter 3.4).
- S. Mustonen: Tilastolliset monimuuttujamenetelmät,(Chapter 1, Helsinki 1995).

Agenda

- Examining one variable
- Examining the relationship between two variables
- 3D visualization
- Visualizing multidimensional data

Examining one variable

- Histogram
- Represents the frequency of occurences within data categories
- one value (for discrete variable)
- an interval (for continuous variable)

Examining one variable

- Stem and leaf diagram (A&B)
- Presents the same graphical information as histogram
- provides also an enumeration of the actual data values

Examining the relationship between two variables

- Scatterplot
- Relationship of two variables

Linear

Non-linear

No correlation

Examining the relationship between two variables

- Boxplot (according A&B)
- Representation of data distribution
- Shows:
- Middle 50% distribution
- Median (skewness)
- Whiskers
- Outliers
- Extreme values

3D visualization

- Good if there are just 3 variables
- Mustonen: “Problems will arise when we should show lots of dimensions at the same time. Spinning 3D-images or stereo image pairs give us no help with them.”

Visualizing multidimensional data

- Scatterplot with varying dots
- Scatterplot matrix
- Multivariate profiles
- Star picture
- Andrews’ Fourier transformations
- Metroglyphs (Anderson)
- Chernoff’s faces

Scatterplot

- Two variables for x- and y-axis
- Other variables can be represented by
- dot size, square size
- height of rectangle
- width of rectangle
- color

Scatterplot matrix

- Also named as Draftsman’s display
- Histograms on diagonal
- Scatterplot on lower portion
- Correlations on upper portion

Scatterplot matrix (cont…)

- Shows relations between each variable pair
- Does not determine common distribution exactly
- A good mean to learn new material
- Helps when finding variable transformations

Scatterplot matrix as rasterplot

- Color level represents the value
- e.g. values are mapped to gray levels 0-255

Multivariate profiles

- A&B: ”The objective of the multivariate profiles is to portray the data in a manner that enables each identification of differences and similarities.”
- Line diagram
- Variables on x-axis
- Scaled (or mapped) values on y-axis

Multivariate profiles (cont…)

- An own diagram for each measurement (or measurement group)

Star picture

- Like multivariate profile, but drawn from a point instead of x-axis
- Vectors have constant angle

Andrews’ Fourier transformations

- D.F. Andrews, 1972.
- Each measurement X = (X1, X2,..., Xp) is represented by the function below, where - < t < .

Andrews’ Fourier transformations (cont…)

- If severeal measurements are put into the same diagram similar measurements are close to each other.
- The distance of curves is the Euklidean distance in p-dim space
- Variables should be ordered by importance

Andrews’ Fourier transformations (cont…)

- Can be drawn also using polar coordinates

Metroglyphs (Andersson)

- Each data vector (X) is symbolically represented by a metroglyph
- Consists of a circle and set of h rays to the h variables of X.
- The lenght of the ray represents the value of variable

Metroglyphs (cont...)

- Normally rays should be placed at easily visualized and remembered positions
- Can be slant in the same direction
- the better way if there is a large number of metrogyphs

Metroglyphs (cont...)

- Theoretically no limit to the number of vectors
- In practice, human eye works most efficiently with no more than 3-7 rays
- Metroglyphs can be put into scatter diagram => removes 2 vectors

Chernoff’s faces

- H. Chernoff, 1973
- Based on the idea that people can detect and remember faces very well
- Variables determine the face features with linear transformation
- Mustonen: "Funny idea, but not used in practice."

Chernoff’s faces (cont…)

- Originally 18 features
- Radius to corner of face OP
- Angle of OP to horizontal
- Vertical size of face OU
- Eccentricity of upper face
- Eccentricity of lower face
- Length of nose
- Vertical position of mouth
- Curvature of mouth 1/R
- Width of mouth

Chernoff’s faces (cont…)

- Face features (cont…)
- Vertical position of eyes
- Separation of eyes
- Slant of eyes
- Eccentricity of eyes
- Size of eyes
- Position of pupils
- Vertical position of eyebrows
- Slant of eyebrows
- Size of eyebrows

Conclusion

- Graphical Examination eases the understanding of variable relationships
- Mustonen: "Even badly designed image is easier to understand than data matrix.”
- "A picture is worth of a thousand words”

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